 We begin a new module here devoted to treatment of monolayers and in this lecture 1, we will try to address the equations of state, aspects of cohesion, repulsion. By now you have some idea about the monolayers and the kind of apparatus required to study those. Langmuir Truff has been mentioned earlier to you and one could make use of measurement of the surface pressure versus area using a Langmuir Truff. So, let us deal with the pi versus A curves, the relations between pi and A, the film pressure against the area, we often call a surface equation of state and the measurements reveal that the pi versus A curves resemble the curve representing osmotic pressure as a function of concentration in solution. When we think about the theory to relate the surface pressure pi to area, where pi is the film pressure which allows for only the kinetic movement in the surface film and we restrict it to two dimensions with kinetic energy of half kT each. This will remind you of the ideal equation of state for gases or that osmotic pressure concentration relation and this simple equation of state would have certain assumptions, this equation to adequately describe experimental facts. So, we will take up these issues as we go along. The first aspect is that this equation does not allow for any interaction among the molecules present in this surface phase, neither cohesion nor repulsion and in that sense it is a limiting law true only when pi tends to 0, when the surface pressure tends to 0 or we could say from here when pi tends to 0, A tends to infinity. So, area available for a surface molecule tending to infinity or number of molecules in the surface approaching 0 under these conditions pi tending to 0 or n tending to 0 or A tending to infinity, the film pressure pi is related to area per molecule A by this simple ideal relation presuming no interaction or you could look at it in the light of external data. When you make measurements on the surface pressure pi versus the area and make a plot like pi versus 10 to the power 4 by A, the external data show these points. So, the actual curve relating pi to A would be what should be running through these data points. However, if you look at the early part of this pi versus reciprocal A, you can have this tangent a straight line with the equation pi equal to k t, pi proportional to inverse of A that is a constant. So, pi equal to k t that is the limiting law that we are talking of when A tends to infinity or n tends to 0 or pi tends to 0. So, our external data bear out this ideal equation for the surface, the surface equation of state pi equal to k t only in this initial portion. And that is understandable when the areas are very large, the molecules in the surface are so far apart that they would have hardly any interaction. These data have been obtained for meristic acid on 100 normal HCl at 17 degree centigrade. So, pi versus this reciprocal A or 10 to the power 4 by A when plotted for film pressures measured for meristic acid on 0.01 normal HCl, give you this curvilinear behavior for which pi A equal to k t is the asymptotic tangent here. Pi is inversely proportional to A proportionally constant would depend on temperature that would be k t giving you pi A equal to k t. So, we see that ideal surface equation of state is valid for very low pressures or very large areas. If the molecular weight of the adsorb or spread film material is not known, we may be able to deduce it and for that we need to write an appropriate form of the ideal surface equation of state valid for n moles. So, if capital N is the number of moles pi A equal to n r t that would be the relevant surface equation of state. And if we look at either this or the previous one, we understand that at large surface pressures pi these equations would predict A value approaching 0. If you take pi on the other side and let pi take a large value, then k t by pi or n r t by pi would approach 0 as pi becomes very large. However, for straight hydrocarbon chains, we should understand that there has to be a limiting area. That limiting area will call A 0 will be of the order of 19.5 to 20 square angstroms. This may remind you of the concept of excluded volume in the van der Waals equation of state for bulk gases wherein we have to allow for the actual volume occupied by molecules of gas. Same way in the surface, when the pressure becomes very large, you can only reduce A to some minimum value per molecule which will correspond to this limiting area roughly given by cross sectional area of the straight hydrocarbon chain. That will be working out to be about 19.5 to 20 angstroms square. Remember this was the number we also ran into when we looked at the concept that if a film at the surface is thinned out, then without breaking into islands, it could only come down to a certain minimum thickness which will correspond to this limiting area. Now, allowing for this, Langmuir proposed that the kinetic energy of molecules in the gaseous film. Remember here, gaseous is being referred to for situations corresponding to very high pressures when the molecules are far apart. In a sense, you can visualize that this is a two-dimensional equivalent of a gas where the molecules are far apart to exert much influence on each other. So, kinetic energy under these conditions could be corresponding to this film pressure pi k. That is a calculated pressure due to kinetic energy of molecule in the film and the limiting area is a 0. So, pi k into a minus a 0 equal to k t or under the conditions where there is no interaction, the total film pressure is pi same as pi k. So, pi into a minus a 0 equal to k t or pi k into a minus a 0 equal to k t, that would be the appropriate equation of state allowing for that limiting area. Now, we do not have that problem when pi or pi k becomes very large, a tends to a 0. We allow for this actual area a 0 occupied by molecules in the surface and when pi is small, we will of course, have very large area per molecule and then in comparison to this large area per molecule a 0 is small. So, we will be able to approximate a minus a 0 by a and therefore from equation 2 we will get pi equal to k t. So, the ideal surface equation of state comes out from this when pi is very small and when pi is very large, a approaches a 0. Now, this equation proposed by Langmuir is often obeyed by electrically neutral films present at interfaces between oil and water. Now, you might have a situation where the monolayer may not be completely mobile. At air oil interface, the assumption that the monolayer molecules are completely mobile is fine, but if you were to think of say water solid interface like water paraffin wax interface, then the monolayer at the interface might not be mobile as at water paraffin oil interface. So, if the molecules forming the monolayer are now held in place at fixed sites on the surface of the solid, we require a different surface equation of state and that is given by Frumkin and Wallmer. Frumkin and Wallmer equation is written as follows, pi equal to k t by a 0, ln of a by a minus a 0 and these two equations, Langmuir equation and Frumkin-Wallmer equations have been tested experimentally for butyric acid adsorbed at oil-water interface and at paraffin wax-water interface and we could see that this equation 3 is actually obeyed at the liquid solid interface. Let me sketch the experimental data which confirm this. Now, we are looking at pi versus 10 to the power 4 by a pi is in milli dynes per centimeter in the range 0 to 150 milli dynes per centimeter. The experimental data show the behavior of this type where pi a equal to k t will be the asymptotic representation. That is how your meristic acid on sodium chloride solution will confirm the ideal surface equation of state. If you take the Frumkin-Wallmer equation, we could look at experimental data for water against paraffin wax and pi now against a will show you experimental data for mobile film and for immobile surface like oil-water interface and for immobile surface like the paraffin wax-water interface and this will be this data set bears out the Frumkin-Wallmer surface equation of state. So, at liquid solid interfaces, the adsorption at fixed sites leads to this kind of surface equation of state equation 3. Now we move on to consider the cohesion. If we take air-water surfaces, we expect they should be cohesion within the film. If the film is like of oleic acid or steric acid, you know from your knowledge of chemistry that there is a slight difference here. The oleic acid will have a double bond in the chain. Nevertheless, at air-water surfaces when steric acid or oleic acid monolayer is adsorbed, because of the van der Waals attraction among hydrocarbon chains, we will have certain cohesive pressure. Cohesion is characterized by cohesive pressure which will crescent as pi s. So, the total surface pressure or film pressure is what is the contribution from the kinetic movement pi k and the cohesion or cohesive pressure pi s. And here pi k is the value that you would get from the Langmuir modification. Pi k is k t by a minus a 0. If pi k is k t by a minus a 0 and cohesion has to contribute to total pressure, it is not difficult to see that this cohesive pressure pi s ought to be negative. So, this is trying to have the chains held closer together, especially at air-water surfaces. You might be wondering why I am emphasizing air-water surfaces again and again. There is a reason, alright. If pi s is negative, what could give an estimate of this pi s? Under conditions where area per molecule is greater than 100 square angstroms, we have an empirical relation for state chain compounds which gives you pi s expectedly minus 400 m, where m is the number of CH2 groups in the chain divided by area to the power 3 by 2. So, this could be adequate empirical equation for the cohesive pressure. Below this area less than 100 square angstroms, the cohesion does not rise as sharply as this equation seems to indicate. But within that limitation of areas greater than 100 angstrom square per molecule, we should be able to write the surface equation of state for the cohering film as follows. Because pi is pi k plus pi s, pi k is pi minus pi s and pi s is that minus 400 m by a to the power 3 by 2, substituting that over here, we get pi k equal to pi plus 400 m by a to the power 3 by 2. And since we have pi k a minus a 0 equal to k t, substituting for pi k, we get this equation. So, allowing for the kinetic movement in the film and cohesion, we have this new equation pi plus 400 m by a to the power 3 by 2 a minus a 0 equal to k t as the appropriate surface equation of state. We next address equation of finding pi s. We go back to air water interfaces. When the films are present at air water interfaces, we do expect cohesion and pi will be equal to pi k plus pi s. Pi k is given by this k t by a minus a 0. So, by measuring the force area curve that is pi versus a relation and knowing pi k is this, we can find out pi s. But what would happen if we think of a situation where our monolayer is instead at an oil water interface? Think of this. This is exaggerated, but we understand there will be this interaction, cohesion among the chain, among the chains of this adsorb molecules. What would happen at oil water interface? I need you to think about this. You will have to make use of a certain concept that we had discussed earlier. d y uncle dent. What is that? d y uncle dent. No, what is the difference between these two? If I call this system kind 1 against this system kind 2, what is the main difference that you see over here? Let us start thinking directly from what is obvious, spot the difference between system 1 and system 2. What is the difference? I have not drawn the film here, but supposing that I have a film here, what is it that is different? Water is same, you have air here and oil here. Air is gaseous, so there are very few molecules. It is an air medium. Oil is a condensed space. So, lots of oil molecules are there over here, which I am not indicating here. The chains are hydrophobic. So, if the chains are hydrophobic, they are oleophilic, oil loving. So, while these chains can come close together here with very few of air molecules, oxygen, nitrogen molecules predominantly being able to make their way here, that is not the situation for oil. Now, you can have oil molecules in between. These chains are oil loving. So, surely these chains are further apart than the nearest neighbors. Here, they can be almost nearest neighbors. Here, they are further apart than the nearest neighbors and the interaction among molecules drops very rapidly with the distance. So, if the oil molecules are getting in between, the cohesion among chains will be drastically reduced. That would mean practically actually in the oil-water interfaces, the monolayers could be safely assumed to have no cohesion. So, going from air-water to oil-water interface, we can remove the cohesion practically completely. So, there is another way here to get pi s. First one was this pi k plus pi s is pi, estimating pi k as k t by a minus is 0, that allows pi s. The other way is compare data for the force area curve at air-water against oil-water interface. At air-water interface, again pi k is k t by a minus is 0 and pi is pi k plus pi s. At oil-water interface, for the reasons given, pi s is 0. So, now, you can use the data on the force area curves or the equation of state for air-water and oil-water interfaces to get the magnitude of pi s. In general, we may think of the broader picture. We might actually have even repulsive pressure. If there is a repulsion among the surface film molecules, the total pressure at air-water interface could be given as pi a w equal to pi k plus pi s plus pi r. Pi r is the component of the surface repulsive pressure and that could originate in the electrical energy of the double layer. The double layer I talked about last time is now capable of producing repulsion in the surface and that surface repulsive pressure is pi r. Considering that monolayer is such that it can exhibit cohesion from the chains, repulsion from the head groups or arising out of that electrical energy of double layer, the total pressure at air-water interface will be some of these three, pi k, pi s and pi r. But the same monolayer, if you visualize that oil-water interface, pi s will be eliminated and you get pi k plus pi r. Then, making measurements at air-water and oil-water interface, we could estimate pi s. The results for octa-dacil tri-methyl ammonium ions show that this actually is true. For neutral films, therefore, we have more or less complete analysis. Let us take it further, where the repulsion will be very significant, that is the case of charged films. Such a film will have a net electrical charge and the result will be increase in pi at any given area per chain. How do we infer the repulsive energy? It is given by Davies equation, which is shown on the next slide. Pi r is given as 6.1 C i to the power 1 by 2 cos hyperbolic sin hyperbolic inverse 134 by E C i to the power 1 by 2 minus 1 at 20 degree centigrade. Or, if the factor A is given as C i to the power 1 by 2 is less than 38, a simpler approximation results, pi r is given by 2 K T by A minus 6.1 C i to the power 1 by 2. Now, what you could do is that is a component corresponding to the repulsive pressure. Pi r is a component of the total pressure. Total pressure is pi k plus pi s plus pi r. Pi r is the repulsive part. Pi s is the cohesive part and pi k is what you would anticipate in absence of any interaction. So, we are taking all these into account. This pi r is due to the repulsion and due to charge. Yes, yeah, yeah. This is because of the repulsive energy corresponding to the charge molecules. Why should it be temperature dependent? The temperature would play a part because you would see that the electrical potential is dependent on temperature. For example, here the interfacial potential is dependent on temperature. Interfacial potential will be a measure of the electrical energy of the double layer. And it is this energy which is responsible for the repulsion among the molecules. So, because the surface or interfacial potential is dependent on temperature, the repulsive pressure pi r will be dependent on temperature. You see the parts of this equation are coming from there, that factor 134 by A c i to the power 1 by 2. This factor is coming from the interfacial energy. Anyway, now for that limited condition of area per chain being less than 38 by c i to the power 1 by 2, you could try simplifying this. This will work out to 2 k T by A minus 6.1 c i to the power 1 by 2. At this point of time, I will let you do a little bit of work. Now, take that pi as pi k plus pi s plus pi r. And you can take pi r as an approximation equal to 2 k T by A minus 6.1 c i to the power 1 by 2. And under that approximation derive the complete, but approximate equation of state for charge films. That you should be do. It is only minor algebra now. So, that should give you that complete, but approximate equation as this equation 9. If you wait to apply this at oil water interface, this second term which is cohesional pressure that should vanish. So, we will exclude this at oil water interface. And this equation has been tested for many systems containing say Lorel sulfonate and sodium chloride solution or sodium Lorel sulfate and no salt in the aqueous phase for alpha bromopamethate on 0.06 normal sodium ions in aqueous solution. Before I go into the slides further that would be probably in the next lecture. I will just ask you to think about something more general. So, supposing pi versus A plots are looked into. You have a reminder of equation of state for say gases, but because now this is a two dimensional equivalent bulk pressure pi being replaced by P being replaced by pi, volume replaced by area per molecule. Unlike for gases and unlike for liquids and solids, we might be able to get a spectrum of behavior which will be differentiated on this diagram itself pi versus A. Think for instance the molecules in the surface film are able to have different degrees of cohesion. If there is no cohesion at all and no repulsion, we will get that pi equal to k t or pi into A minus A 0 equal to k t. In light of experimental measurements and you already know what happens to steric acid films or monolayers when they are given very large area, the kinds of experiments Lord really had done. If you give too much area to such strongly cohering films, the monolayer breaks up into islands, but within any island there is still considerable inter chain cohesion. In fact, the reason we got these islands as a result of breakup of the monolayer is a reflection of the fact that there is a strong very strong cohesion among the chains. You could look at the other extreme. Supposing we had such a film of steric acid and you were to restrict it in the surface between two barriers, one which is movable and the other which is fixed. You could have a Langmuir trough, you could have a monolayer and then you have two surface barriers, one is fixed and other can be moved. You can actually compress the surface film and at every area available per molecule, you can record the film pressure. When the molecules are close together and area per chain or per molecule is of the order of a 0 that rough 20 angstrom square, what do you expect to see? If you try to decrease the area further, what do you expect will happen to the film pressure or pi will rise. Area cannot decrease because now we are approaching limiting area, pressure will start increasing and pressure will increase very rapidly. So, you can visualize that pi versus a curve will have a very steep increase. Contrast this against the situation where there is no interaction, lots of area no interaction, there the decrease in area will result in gradual increase in the film pressure. So, these are two extremes we expect. So, the first behavior pi equal to k t that will be somewhat like gas like behavior. The monolayer behaves as if it is a two dimensional gaseous material in the film whereas, steric acid will be representative of condensed phase offering very steep rise in pressure pi with decrease in area and we distinguish the two, one as a gaseous monolayer and this as the condensed monolayer. In between you may get a intermediate sensitivity of pi to a that we will term as liquid or liquid expanded monolayer. So, now you see that in the two dimensional equivalent in the surface phase, the surface equation of state brings out a parallel to what is for gas, what is for liquid and what is for solid. But all of this is only the difference in the nature of the molecules forming the surface film. So, different states of matter are analogously represented in the surface. I will like to expand on this concept in more quantitative terms as we go further and let us see how far we can take it. We will need to use some of the thermodynamics knowledge in the process and I would like to do it such that it is not a only verbal qualitative statement, but there are quantitative calculations supporting these conclusions. So, we will stop here for today.